Formules pour les nombres premiers

Remarques sur le premier cas du théorème de Fermat sur les corps de nombres

The first case of Fermat's Last Theorem for a prime exponent $p$ can sometimes be proved using the existence of local obstructions. In 1823, Sophie Germain has obtained an important result in this direction by establishing that, if $2p+1$ is a prime number, the first case of Fermat's Last Theorem is true for $p$. In this paper, we investigate such obstructions over number fields. We obtain analogous results on Sophie Germain type criteria, for imaginary quadratic fields. Furthermore, extending a well known statement over ${\bf Q}$, we give an easily testable condition which allows occasionally…

Équation de Fermat et nombres premiers inertes

Let $K$ be a number field and $p$ a prime number $\geq 5$. Let us denote by $μ_p$ the group of the $p$th roots of unity. We define $p$ to be $K$-regular if $p$ does not divide the class number of the field $K(μ_p)$. Under the assumption that $p$ is $K$-regular and inert in $K$, we establish the second case of Fermat's Last Theorem over $K$ for the exponent $p$. We use in the proof classical arguments, as well as Faltings' theorem stating that a curve of genus at least two over $K$ has a finite number of $K$-rational points. Moreover, if $K$ is an imaginary quadratic field, other than ${\bf Q}(…

从素数到 Möbius 函数的显式转换

Conversions explicites des nombres premiers vers la fonction de Möbius

本文改进 Möbius 函数部分和的显式上界，在两个不同阈值下分别给出常数 160383 与 180194 的新估计。

Nouvelles conditions pour l'inexistence des nombres parfaits impairs

We will show the two following results: If there existe an odd perfect number $n$ of prime decomposition $n=p_1^{α_1} \ldots p_k^{α_k}q^β$, where the $α_i$ are even, the $β$ are odd and $q \equiv 5 \mod 8$. Then there is at least one $p_i$, $1 \leq i \leq k$ that is not a square in $\mathbb{Z}/q\mathbb{Z}$. More precisely there is an odd number of $p_i$ that are not squares in $\mathbb{Z}/q\mathbb{Z}$. If there exist an odd perfect number $n$ of prime decomposition $n=p_1^{α_1} \ldots p_k^{α_k}q^β$, where the $α_i$ are even, the $β$ are odd and $p_{k+1} \equiv {1\mod 8}$. Then at least one $p_…

Compter (rapidement) le nombre de solutions d'équations dans les corps finis

The number of solutions in finite fields of a system of polynomial equations obeys a very strong regularity, reflected for example by the rationality of the zeta function of an algebraic variety defined over a finite field, or the modularity of Hasse-Weil's $L$-function of an elliptic curve over $\Q$. Since two decades, efficient methods have been invented to compute effectively this number of solutions, notably in view of cryptographic applications. This exposé presents some of these methods, generally relying on the use of Lefshetz's trace formula in an adequate cohomology theory and discuss…

Formes quadratiques de discriminants emboîtés

Quadratic forms with embedded discriminants. Integral binary quadratic forms have multiple applications, for example in factorization or cryptography. The Nice family of cryptographic systems makes use of quadratic forms with different discriminants $\pm p$, and $\pm pq^2$ where $p$, $q$ are large primes. This paper shows the precise links between forms with $D$ discriminant and forms with $Df^2$ discriminant, which are crucial in the analysis of the systems Nice and theirs attacks. We also introduce the notion of semi-equivalence of binary quadratic forms, and give some characterizations of s…